4 PRAMOD N. ACHAR

Proof. (1) Suppose V = 0, and let m be the smallest integer such that

Hm(V

) = 0. Then, for any object X ∈ V n1 ∗ · · · ∗ V nk , it follows that the

map

Hm(V

n1 ) →

Hm(X)

is injective. But if X = 0, this contradicts the as-

sumption that

Hm(V

) = 0.

(2) The argument given for part (1) shows that

Hi(V

) = 0 for i 0, and that

the map H0(V

) →

H0(k)

∼

= k is injective. Let Y ∈ V n1∗···∗ V nk be such that

there is a distinguished triangle V → k → Y →. If H0(V

) = 0, it would follow that

H0(Y ) = 0, leading to a contradiction with the fact that H0(k) = 0, so it must be

that H0(V )

∼

=

k. We then see from that distinguished triangle that

Hi(V

)

∼

=

Hi−1(Y

) for all i ≥ 1.

Because all the ni are strictly positive, it follows from the fact that

H0(V

)

∼

= k

that

H0(Y

) is concentrated in strictly positive degrees, and hence so is

H1(V

).

Thereafter, we proceed by induction on i: if

Hi(V

) is concentrated in strictly

positive degrees, so is

Hi(Y

), and therefore so is

Hi+1(V

).

2.3. Quasi-hereditary categories. Let S be a set equipped with a partial

order ≤. Assume that every principal lower set is ﬁnite, i.e., that

(2.1) For all s ∈ S, the set {t ∈ S | t ≤ s} is ﬁnite.

Let A be a ﬁnite-length abelian category, and assume that one of the following

holds:

• “Ungraded case”: There is a ﬁxed bijection Irr(A)

∼

= S.

• “Graded case”: A is equipped with a Tate twist, and there is a ﬁxed

bijection Irr(A)

∼

=

S ×Z with the property that for a simple object L ∈ A,

L corresponds to (s, n) if and only if L 1 corresponds to (s, n + 1).

In the ungraded case, choose a representative simple object Σs for each s ∈ S,

and let

(≤s)A

(resp.

(s)A)

be the Serre subcategory of A generated by all simple

objects Σt

with t ≤ s (resp. t s).

In the graded case, let Σs

denote a representative simple object corresponding

to (s, 0) ∈ S × Z. In this case, (≤s)A (resp. (s)A) denotes the Serre subcategory of

A generated by all simple objects Σt n with t ≤ s (resp. t s) and n ∈ Z. More

generally, for any subset Ξ ⊂ S × Z, we let ΞA denote the Serre subcategory of A

generated by the Σt n with (t, n) ∈ Ξ.

In the sequel, we will focus mostly on the graded case. With the above notation

in place, the corresponding deﬁnitions and statements for the ungraded cases can

usually be obtained simply by omitting Tate twists and by changing “Hom” and

“Ext” to “Hom” and “Ext,” respectively. For instance, it is left to the reader to

formulate the ungraded version of the following deﬁnition.

Definition 2.3. A category A as above is said to be graded quasi-hereditary

if for each s ∈ S, there is:

(1) an object Δs and a surjective map φs : Δs Σs such that

ker φs ∈

(s)A

and Hom(Δs, Σt) =

Ext1(Δs,

Σt) = 0 if t s.

(2) an object ∇s and an injective map ψs : Σs → ∇s such that

cok

ψs

∈

(s)A

and Hom(Σt,

∇s)

=

Ext1(Σt, ∇s)

= 0 if t s.

4